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Journal of Convex Analysis 15 (2008), No. 4, 803--818
Copyright Heldermann Verlag 2008

On Semicontinuity of Convex-Valued Multifunctions and Cesari's Property (Q)

Andreas Löhne
Institut für Mathematik, Martin-Luther-Universität, Theodor-Lieser-Straße 5, 06099 Halle, Germany


\newcommand{\R}{\mathbb{R}} We investigate two types of semicontinuity for set-valued maps, Painlev\'{e}-Kuratowski semicontinuity and Cesari's property (Q). It is shown that, in the context of convex-valued maps, the concepts related to Cesari's property (Q) have better properties than the concepts in the sense of Painlev\'{e}-Kuratowski. In particular we give a characterization of Cesari's property (Q) in terms of upper semicontinuity of a family of scalar functions $\sigma_{f(\,\cdot\,)}(y^*) \colon X \to \overline\R$, where $\sigma_{f(x)} \colon Y^*\to \overline\R$ is the support function of the set $f(x)$. We compare both types of semicontinuity and show their coincidence in special cases.

Keywords: Semi-continuity, set-valued maps, property (Q).

MSC: 47H04,58C07

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